Index: The Book of Statistical ProofsStatistical Models ▷ Univariate normal data ▷ Multiple linear regression ▷ t-test for single regressor

Theorem: Consider a linear regression model

\[\label{eq:mlr} y = X\beta + \varepsilon, \; \varepsilon \sim \mathcal{N}(0, \sigma^2 V)\]

using the $n \times p$ design matrix $X$ and the parameter estimates

\[\label{eq:mlr-est} \begin{split} \hat{\beta} &= (X^\mathrm{T} V^{-1} X)^{-1} X^\mathrm{T} V^{-1} y \\ \hat{\sigma}^2 &= \frac{1}{n-p} (y-X\hat{\beta})^\mathrm{T} V^{-1} (y-X\hat{\beta}) \; . \end{split}\]

Then, the test statistic

\[\label{eq:mlr-t-single} t_j = \frac{\hat{\beta}_j}{\sqrt{\left( \hat{\varepsilon}^\mathrm{T} V^{-1} \hat{\varepsilon} \right)/(n-p) \; \sigma_{jj}}}\]

with the $n \times 1$ vector of residuals

\[\label{eq:mlr-eps-est} \hat{\varepsilon} = y - X\hat{\beta}\]

and $\sigma_{jj}$ equal to the $j$-th diagonal element of the parameter covariance matrix

\[\label{eq:mlr-t-single-sig} \sigma_{jj} = \left[ \left( X^\mathrm{T} V^{-1} X \right)^{-1} \right]_{jj}\]

follows a t-distribution

\[\label{eq:mlr-t-single-dist} t_j \sim \mathrm{t}(n-p)\]

under the null hypothesis that the $j$-th regression coefficient is zero:

\[\label{eq:mlr-t-single-h0} H_0: \; \beta_j = 0 \; .\]

Proof: This is a special case of the contrast-based t-test for multiple linear regression based on the following t-statistic:

\[\label{eq:mlr-t} t = \frac{c^\mathrm{T} \hat{\beta}}{\sqrt{\hat{\sigma}^2 c^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} c}} \sim \mathrm{t}(n-p) \; .\]

In this special case, the contrast vector is equal to the $j$-th elementary vector $e_j$ (a $p \times 1$ vector of zeros, with a single $1$ in the $j$-th entry)

\[\label{eq:mlr-t-single-con} c = e_j = \left[ 0, \ldots, 0, 1, 0, \ldots, 0 \right]^\mathrm{T} \; ,\]

such that the null hypothesis is given by

\[\label{eq:mlr-t-single-h0-qed} H_0: \; c^\mathrm{T} \beta = e_j^\mathrm{T} \beta = \beta_j = 0\]

and the test statistic becomes

\[\label{eq:mlr-t-single-qed} \begin{split} t_j &= \frac{e_j^\mathrm{T} \hat{\beta}}{\sqrt{\hat{\sigma}^2 e_j^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} e_j}} \\ &= \frac{\left[ 0, \ldots, 0, 1, 0, \ldots, 0 \right] \left[ \hat{\beta}_1, \ldots, \beta_{j-1}, \beta_j, \beta_{j+1}, \ldots, \hat{\beta}_p \right]^\mathrm{T}}{\sqrt{\frac{1}{n-p} (y-X\hat{\beta})^\mathrm{T} V^{-1} (y-X\hat{\beta}) \left[ 0, \ldots, 1, \ldots, 0 \right] (X^\mathrm{T} V^{-1} X)^{-1} \left[ 0, \ldots, 1, \ldots, 0 \right]^\mathrm{T}}} \\ &= \frac{\hat{\beta}_j}{\sqrt{\frac{1}{n-p} \left( \hat{\varepsilon}^\mathrm{T} V^{-1} \hat{\varepsilon} \right) \left[ \left( X^\mathrm{T} V^{-1} X \right)^{-1} \right]_{jj}}} \\ &= \frac{\hat{\beta}_j}{\sqrt{\left( \hat{\varepsilon}^\mathrm{T} V^{-1} \hat{\varepsilon} \right)/(n-p) \; \sigma_{jj}}} \; . \end{split}\]

Metadata: ID: P450 | shortcut: mlr-tsingle | author: JoramSoch | date: 2024-05-03, 14:37.